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<span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="scmfsa6b.html#V2" title="SCMFSA6B:attr.2">parahalting</a>  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b2">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st  ex <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> or <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) holds <br/>(  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Olive" title="b2">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Olive" title="b2">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>   st  ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> or <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) holds <br/>(  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Olive" title="b1">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Olive" title="b1">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> or <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) implies (  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font> ) )</span><br/>





<a NAME="E1:57"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
( ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font> ) &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#T24" title="SCMFSA7B:th.24">SCMFSA7B:24</a>, <a class="ref" href="scmfsa7b.html#T25" title="SCMFSA7B:th.25">SCMFSA7B:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:57"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> </span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds <br/> ( ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> or <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font> )</span><br/>

<span class="kw">hence </span><a NAME="E3:57"/>
(  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <font color="Maroon" title="c5">s</font> &amp;  <a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <font color="Maroon" title="c5">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="scmfsa_9.html#T53" target="_self" title="SCMFSA_9:th.53">Th53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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